We have our own instance of Stack Overflow web app which we access using https://stackoverflow.ourcompany.com.
Can I configure the mobile app to connect to that?
No, this is not possible. You cannot configure anything in the app, all URLs are hardcoded to the Stack Exchange API and a few
mobile.stackexchange.com endpoints. The apps don't even work with Stack Overflow for Teams, even though the API has some support for it.
Since development on the apps is frozen, you're better off building something yourself. I'm not sure if Stack Overflow for Enterprise already supports the new responsive design, which works pretty well on mobile. All you'd need is a small wrapper/container for push notifications, and you have a decent app. According to the product splash page, it comes with an API you could use for this.